Contributions to Algebra and Geometry Volume 50 (2009), No. 2, 433-441.
A Note on Ovals and their Evolutoides
Marco Hamann
Institute for Geometry, Dresden University of Technology Zellescher Weg 12–14, Willersbau B107, D-1062 Dresden, Germany
e-mail: [email protected]
Abstract. In this paper we consider curves related to ovals, which can uniformly be generated by certain families of secants with respect to an arbitrary given oval. If e. g. the generating secants connect the points of contact of parallel tangents of the given oval then they envelop a curve, which contains information about the oval analogously to its evolute for example. Those curves are treated in different context in [2] and [6] too.
In this article several properties can be treated into detail and extended respectively. In particular with help of this curve a global property to any evolutoide of the given oval can be shown, which is invariant under (regular) affine transformations. It generalizes a corresponding result about the evolute of the oval, see [2].
MSC 2000: 53A04, 51M04, 51N20
Keywords: evolute, evolutoide, kinematics, minimal curve, oval, certain secants
1. Introduction
Curves related to ovals have been studied in many papers by mathematicians and engineers, too. Many questions were initiated by technical problems and have applications in engineering respectively. For example the study of isoptic curves is associated to cam mechanisms. As is generally known those curves are the loci, where the contour of the given oval is seen under fixed angle measure.
Generalizations of isoptic curves are well-known: for example dually defined curves are generated by families of chords of equal length with respect to the given oval.
They seem to be of some interest in current research too, see [4].
0138-4821/93 $ 2.50 c 2009 Heldermann Verlag
On the other hand in context to the generation of these curves by secants it seems to be natural to consider curves generated by a one-parameter family of certain secants with respect to an ovalk. There exist numerous classical examples of it, in particular the evolute eassociated to an oval k generated as envelope by the family of curve normals ofk. As is generally known the definition of an evolute e can be generalized onto evolutoides eα, if the normals of k are replaced by a one-parameter set of lines intersectingk isogonally. The constant angle (measure) of the intersection of the oriented line and the tangent in a point of k is denoted byα. For example those one-parameter sets of certain lines occur as ground view of horizontal tangents at points P ∈ k of surfaces of constant slop, where k is itself a curve of constant slop.
In different context curves related to ovals are treated in [2]. The so-called minimal curve of a given ovalk is defined as locus of certain centers Z at secants s, which connect the points of contact of parallel tangents of k. In a point Z the curvature of a certain convex curve kZ related to k is stationary (with respect to Z ∈s) and has an absolute minimum. It is shown in [2] that the set of points Z can be generated as envelope of the family {s}.
However the minimal curve might have further properties in context to the given ovalkand its evolutoideseα: in this article several properties can be treated into detail and extended respectively. In particular with help of this curve a global property to any evolutoide of the given oval can be shown, which is invariant under (regular) affine transformations. It generalizes a corresponding result about the evolute of the oval, see [2]. Finally when placing the emphasis on geometric optic minimal curves admit a generation by certain secants swith respect to an oval k.
2. Definition of a midenvelope
Let k be an oval in the Euclidean plane E2, that is a C2-smooth strictly con- vex curve. Using a Cartesian coordinate system the curve k has non vanishing curvature κ. Without loss of generality the origin O of the coordinate system can be chosen as an inner point of the edge k. Then k is enveloped of all sup- porting (tangent) lines xcost+ysint−p(t) = 0 whereby the support function1 p: [0,2π)→R+. Using complex numbers the oval k is parametrized by
z(t) = p(t)eit+ ˙p(t)ieit, z(t)∈C1 (1) whereby t ∈[0,2π)⊂R. The dot in (1) stands for the derivation with respect to the real parameter t.
From the definition ofk follows that for everyt ∈[0,2π) there exists a unique valuet+π, so that the distinct supporting lines l(t) and l(t+π) with respect tok are parallel.2 The points of contact are denoted by z(t) andz(t+π) respectively.
Hence fort ∈[0,2π) the oriented lines(t) belonging to the ordered pair (z(t), z(t+
π)) is well defined. In the following we investigate the family of secants s(t)
1Sometimes the support function is defined over the Gaussian image of a curve, see e. g. [1].
2More precisely it should be denoted (t+π) (mod 2π) instead oft+π.
through z(t) andz(t+π) for all t. Those secants can be geometrically interpreted as lines intersecting the (tangents of the) curve k in complementary angles α(t) and α(t+π). But notice the measure of α(t) changes generally when changing t.
Definition 1. Let k be an oval parametrized by (1). Then the family of oriented lines s(t) belonging to the ordered pairs (z(t), z(t+π)) of k for t∈ [0,2π) deter- mines a set of common points and a curve crespectively, which is the envelope of this family of certain secants of k. They should be named shortly diameters s(t) and midenvelope c with respect to k.3
Example 1. As is generally known the family of diameters s(t) with respect to a curve k of constant width are the double normals of k. Hence the midenvelope cis the evolute of k. In particular if k is a circle then ccoincides with the center of k,s(t) are diameters of k.
To compute the equation for the midenvelopecand investigate respectively wheth- er c contains common points of the diameters s(t) we first have to compute the equation of s(t) as the line connecting two distinct points z(t) and z(t+π) of k.
Using the notation in (1) we obtain 0 = a(t) sint+ ˙a(t) cost
x+ −a(t) cost+ ˙a(t) sint
y−b(t) (2) with a(t) = p(t) +p(t+π) and b(t) = p(t) ˙p(t+π)−p(t)p(t˙ +π). As a necessary condition thatcis an envelope with respect to{s(t)}, we solve the linear equation system given by (2) and the subsequent equation (3), which is determined by derivation of left and right hand side of (2) with respect to t.
0 = cost a(t) + ¨a(t)
x+ sint a(t) + ¨a(t)
y−b(t).˙ (3)
Obviously (2) and (3) do not change iftis replaced byt+π, hence the solutions at t and t+π coincide. Ifb(t) = 0 then the linear equation system is homogeneous.
Since the radius of curvature of the ovalk is greater than 0 for anyt, furthermore evidentlya(t)6= 0, the unique solution of the homogeneous linear equation system is determined by zc = 0.4 It is easy to verify that b(t) = 0 if and only if the function f : [0, π) → R+ with t 7→ p(t)/p(t +π) has a local extremum and a point of inflection with horizontal stationery tangent at t respectively or is constant. Therefore the conditionb(t) = 0 for everyt∈[0,2π) admits a geometric interpretation with respect to the given oval: k is then centrally symmetric. In this case the family of diameterss(t) is a pencil of lines throughzc = 0.
Generaly the solution of the linear equation system given by (2) and (3) gives a parametrization of c. We excluded the case the solution zc is fixed for every
3The pairs (l(t), l(t+π)) of parallel supporting lines, the family of diameters s(t) and the midenvelopec are affine geometric properties. They are studied in different context in [2] too.
With respect to the definition in [2] we will use different names fors(t) andc.
4The radiusr of curvature is determined by the support function, thusr=p(t) + ¨p(t)>0, see e. g. [5].
t ∈ [0,2π) and equivalent, k is a centrally symmetric oval respectively.5 Using complex numbers we get
zc(t) = a(a+ ¨a)−1
abe˙ it+ ( ˙ab˙ −b(a+ ¨a))ieit
(t) (4)
where t ∈ [0, t) (mod π). Obviously the denominator in (4) cannot vanish in [0,2π).
To obtain a proposition, when a midenvelope c is regular at t, we consider the tangent vector to c at t. Equivalent to the zero tangent vector the condition
|z˙c(t)|= 0 is fulfilled. Provided that k is an oval we get 0 = (a+ ¨a)(ab−a˙b˙ +a¨b+b¨a)−ab( ˙˙ a+ ˙¨a)
(t) (5)
and replacing a and b and their derivations by the support function p 0 = p(t+π) + ¨p(t+π)
˙
p(t) + ˙¨p(t)
− p(t˙ +π) + ˙¨p(t+π)
p(t) + ¨p(t) . (6) Note the functions a, b and the support function p in (5) and (6) respectively are assumed to be of class C3. Evidently the second condition is geometrically interpreted by the curvature radii ofk attand t+π: as noticed before the radius of curvature of k at t is given by r := p(t) + ¨p(t), hence the midenvelope c with respect to k is singular at t if and only if the function g : [0, π) → R+ with t 7→r(t)/r(t+π) has a local extremum and a point of inflection with horizontal stationery tangent at t respectively or is constant.
Proposition 1. Let k be an arbitrary oval parametrized by (1) and c the miden- velope with respect to k parametrized by (4). Furthermore the support function p in (1) is of classC3. Then c is singular at t (mod π) if and only if (6) is fulfilled and the function g : [0, π) → R+ with t 7→ r(t)/r(t+π) has a local extremum and a point of inflection with horizontal stationary tangent at t respectively or is constant.
Example 2. Obviously for centrally symmetric ovals r(t)/r(t+π) = 1 for every t ∈ [0,2π). The midenvelope c is determined by the center of k and therefore singular at any t (mod π).
Example 3. The midenvelope c belonging to a curve k of constant width is de- termined by its evolutee. Thereforea(t) =p(t)+p(t+π) =r(t)+r(t+π) = const.
It is easy to verify, if k is different from a circle, the function g : [0, π)→R+ has local extrema exactly at thoset, wherer(t) is extremal too. This is not surprising, because the evolute of a smooth curve with nonvanishing curvature is generally singular at those t, at which k has vertices.
As is generally known the curvature of a smooth plane curve, which is parametriz- ed by z :I →Cwith z(I)∈C2(I), can be computed by
κ(t) = z(t)¨¯˙ z(t)−z(t)¯˙ z(t)¨
2i|z(t)|˙ 3 . (7)
5In the case of a fixed solution we easily get b(t) = 0 for all t ∈ [0,2π) by changing the coordinate system.
Using (4) and (5) the curvature of the midenvelope c with respect to an oval k can be computed with reference to a, b and their derivations by
κc(t) = a4(a+ ¨a)4b a2+ ˙a232
(a+ ¨a)(ab−a˙b˙+a¨b+b¨a)−ab( ˙˙ a+ ˙¨a)2(t). (8) It easily can be verified the expression a2 + ˙a212
(t) in (8) admits an geometric interpretation as the length of the secant segment between z(t) andz(t+π). For a strictly convex smooth curve k it never vanishes.
3. Relation to the evolute of k
As shown in Section 2 the midenvelopecwith respect to an oval k coincides with the evolute e of k, if and only if k is a curve of constant width. In general the points ofcandeattandt+πrespectively are different from each other. It seems natural to ask whether they have a certain position to each other, compare also [2].
Let k be an oval parametrized by (1). Using complex numbers the evolute e of k can be described by equation
ze(t) = ˙p(t)ieit−p(t)e¨ it. (9) We ask whether the points ze(t), ze(t+π) and zc(t) are collinear. Therefore we compute an equation of the (oriented) line g belonging to the (ordered) pair (ze(t), ze(t+π)) and verify a condition wheng containszc(t). Using the notation in (9) we obtain
ze(t+π) =−p(t˙ +π)ieit+ ¨p(t+π)eit (10) and further an equation of g(t)
0 = ˙a(t) cost−¨a(t) sint
x+ ˙a(t) sint+ ¨a(t) cost
y−d(t) (11) with d(t) = ˙p(t)¨p(t+π)−p(t) ˙¨ p(t+π). Obviously (11) does not change if t is replaced by t+π. Hence the solutions at t and t+π coincide. Furthermore g(t) is undefined at t if and only if ze(t) and ze(t+π) coincide. Using (11) we obtain
˙
a= 0 and ¨a= 0. In particular these conditions are fulfilled if the width ofk (with respect to a certain direction) is stationary.
Verifying a condition, when g(t) containszc(t) using the representations (11) and (4), we obtain a global property of cwith respect to an arbitrary oval k.
Proposition 2. Let k be an oval and e its evolute with parameter representa- tions (1) and (9) respectively. Furthermore the midenvelopec with respect to k is parametrized by (4). If ze(t) and ze(t+π) at t ∈ [0, π) are different from each other the connecting line g(t) is well defined and contains the associated point zc(t), i.e. the points ze(t), ze(t+π) and zc(t) are collinear. Otherwise those three points coincide at t∈[0, π).6
6Proposition 2 is similarly elaborated in [2]. With regard to a consistent argumentation and an extension of Proposition 2 for example we have decided to note it down.
The coincidence of the pointsze(t), ze(t+π) and zc(t) in Proposition 2 can easily verified: If ze(t) = ze(t+π) or equivalent ˙a = 0 and ¨a = 0, then the diameter s(t) is a double normal of k att and the width ofk is stationary. Hence the point zc(t) coincide with ze(t) =ze(t+π), compare Example 1.
Corollary 3. If the points ze(t), ze(t+π) and zc(t) are collinear (and do not coincide) at t ∈ [0, π) then zc(t) divides the line segments given by ze(t) and ze(t +π) as well as z(t) and z(t +π) at the same ratio r(t)/r(t +π) of the curvature radii of k at t and t+π.7 Otherwise if those three points coincide at t ∈ [0, π) then the statement is evidently true for the line segment given by z(t) and z(t+π).
Note Corollary 3 can be verified using the theorem on intersecting lines: Let k be an oval parametrized by (1). Then the normals n(t) and n(t+π) to k at t and t+π are parallel to each other. If ze(t) andze(t+π) do not coincide the line g(t) connecting ze(t) and ze(t+π) contains zc(t) by Proposition 2. The points z(t), z(t+π) and zc(t) are collinear by Definition 1. Certainly g(t) is different from s(t).
As a conclusion of Proposition 2 and Corollary 3 we get a global property on the locus of a midenvelope cwith respect to an arbitrary oval k.
Corollary 4. Let k be an oval parametrized by (1), further c the midenvelope with respect to k parametrized by (4). Since k is a closed strictly convex curve, furthermore the curvature κ > 0, the midenvelope c with respect to k lies in the interior of k.
4. Evolutoides
Let k be an oval and c the midenvelope with respect to k. According to [8] the envelopeeαof a one-parameter set of oriented lineslα(t), which isogonally intersect k, is calledevolutoide. The constant angle measure](l(t), lα(t)) of intersection at any t ∈[0,2π) is denoted byα. In the casesα = 0 andα = π2 we speak of k and its evoluteerespectively. Since for these curves the lines s(t) andg(t) connecting z(t), z(t+π) and ze(t), ze(t+π) both contain zc(t) at every t ∈ [0, π), it seems natural to ask whether for α∈[0, π) (α 6= 0,π2) exist evolutoides eα of k with the same property with respect to c.
First we compute a parametrization of the evolutoides eα taking the same methods used in Section 2. Using the representation in (1) we obtain an equation for the line lα(t)
0 = xcos(t−α) +ysin(t−α)− p(t) cosα−p(t) sin˙ α
(12) where t ∈[0,2π) and α is fixed. As a necessary condition that eα is an envelope with respect to {lα(t)}, we solve the linear equation system given by (12) and
7More preciselyzc(t) divide the line segments at their interior.
the subsequent equation (13), which is determined by derivation of left and right hand side of (12) with respect to t.
0 = −xsin (t−α) +ycos (t−α)− p(t) cos˙ α−p(t) sin¨ α
(13) The solution of the equation system generally gives a parametrization of the evo- lutoideseα. Using complex numbers we obtain
ze(t, α) =p(t)ei(t−α)cosα+ ˙p(t)ieit−p(t)ie¨ i(t−α)sinα. (14) Obviously, the parametrization (14) describes the ovalk and its evolutee respec- tively if and only if α= 0 andα= π2 respectively, compare (1) and (9). Therefore the lines connecting ze(t,0), ze(t+π,0) and ze(t,π2), ze(t+π, π2) are determined bys(t) and g(t), further both contain zc(t).
Now we ask whether for α ∈ [0, π) (α 6= 0,π2) exist evolutoides eα of k with the same property with respect to c. Therefore we compute an equation of the line g(t, α) connecting the points ze(t, α) and ze(t+π, α). g(t, α) is well defined as long as those points differ from each other. Using the notation as before we get
0 = a(t) sin (t−α) cosα+ ˙a(t) cost−¨a(t) cos (t−α) sinα x
− a(t) cos (t−α) cosα−a(t) sin˙ t+ ¨a(t) sin (t−α) sinα y
− b(t) cos2α−b(t) cos˙ αsinα+d(t) sin2α .
(15)
With help of the previous equation we can specify a condition, whether the points ze(t, α),ze(t+π, α) and zc(t) are collinear: For a chosenα∈[0, π) the line g(t, α) connectingze(t, α) andze(t+π, α) containszc(t) if and only if the parametrization (4) of c fulfills (15) at t. Based on this condition we obtain an generalization of Proposition 2.
Proposition 5. Let k be an oval and c its midenvelope parametrized by (1) and (4) respectively. Furthermore the family of evolutoides eα with respect to k is de- noted by {eα|α∈[0, π)}. Anyeα admits a parametrization by(14). Ifze(t, α)and ze(t+π, α) at t ∈ [0, π) are different from each other, the connecting line g(t, α) is well defined and contains the associated point zc(t), i.e. the points ze(t, α), ze(t+π, α) and zc(t) are collinear.
Note the points ze(t, α) and ze(t+π, α) coincide if and only if |ze(t, α)−ze(t+ π, α)|= 0. Using (14) we obtain
0 =a2(t) cos2α+ ˙a2(t) + ¨a2(t) sin2α−2a(t) ˙a(t) sinα−2 ˙a(t)¨a(t) cosα (16) which is evidently fulfilled if the width of k is stationary and α = π2, compare Examples 1 and 3.
In the case the points ze(t, α), ze(t+π, α) and zc(t) are collinear (and do not coincide) we obtain a generalization of Corollary 3, which analogously can be verified.
Corollary 6. If the points ze(t, α), ze(t+π, α) and zc(t) are collinear (and do not coincide) at t ∈ [0, π) then zc(t) divides the line segments given by ze(t, α) and ze(t+π, α) as well as z(t)and z(t+π) at the same ratio r(t)/r(t+π) of the curvature radii of k at t and t+π.
Verifying Corollary 6 the caseg(t, α) = g(t+π, α) = s(t) has to be considered for its own: It is a matter of common knowledge for a continuous motion of a moving 2-frame along a plane curve k ∈ C2 that the instantaneous center of this motion coincides with the center of curvature with respect tok, see [7]. Furthermore the normal of a kinematically generated curve at t contains the instantaneous center at t. Referring to the current corollary the locus of any points ze(t, α) at t with α∈[0, π) is a circle with diameter [z(t), ze(t)]. Hence the statement in Corollary 6 evidently continues to be valid ifg(t, α) =g(t+π, α) =s(t), because the functions α→ |ze(t, α)−zc(t)| and α → |ze(t+π, α)−zc(t)| are continuous.
Figure 1. Midenvelope cand evolutoide eα with respect to an oval k Example 4. In Figure 1 a given ovalk and its midenvelope care depicted. The support function p of k is given by p(t) := cos (nt) +m with n = 2l+ 1, l ∈Z+, further m ∈R+ and m > n2−1, see [3]. Obviouslya(t) :=p(t) +p(t+π) = 2m at anyt∈[0,2π) and thereforek is a curve of constant width: cand the evolute e ofk coincide. Further on the evolutoidee(α0) and the loci of points{ze(t0, α)|α∈ [0, π)} and {ze(t0+π, α)|α ∈ [0, π)} are depicted too. The figure visualizes the statements in Proposition 5 and Corollary 6.
5. Conclusion
As conclusions of Proposition 5 and Corollary 6 we can give geometric interpre- tations on the points of the midenvelope c with respect to corresponding points of any evolutoide eα of an arbitrary oval k.
Since the connecting lines of zc(t) and ze(t, α) at t are contained in a pencil with carrier zc(t), furthermore the induced ratios equal r(t)/r(t+π) at any α ∈ [0, π), we can treat all corresponding points ze(t, α), ze(t+π, α) as preimage and image respectively of a dilatation σ(t) given by the center zc(t) and the fixed ratio r(t)/r(t +π). In particular the point z(t), its tangent l(t) and center of curvatureze(t) of k att are mapped by σ(t) onto z(t+π),l(t+π) and ze(t+π) respectively. Hence kσ(t) is in 2nd order contact tok inz(t+π). Furthermore any pair ze(t, α), lα(t)
withα∈[0, π) is mapped onto ze(t+π, α), lα(t+π)
byσ(t).
When t passes through [0,2π) the midenvelope c can therefore be treated as the locus of all centers of dilatations σ(t) as described before. Note there are evidently alternative ways to define contact with respect to k at arbitrary (t1, t2) ∈ [0,2π)×[0,2π) by similarities. Otherwise the pairs (t1, t2) = (t, t+π) with t ∈ [0, π) and therefore c are geometrically determined by k. In particular for central symmetric ovals the midenvelopecis given by the center of symmetry, compare Example 2. In addition in case of an ellipse k the evolutoides eα are affine parastroides, hence central symmetric (with centre c), see [8].
With help of the midenvelope c we can finally simplify the construction of evolutoides eα with respect to an oval k.
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Received July 29, 2008
